UsefulLinks
Mathematics
Combinatorics
1. Introduction to Combinatorics
2. Fundamental Counting Principles
3. Permutations
4. Combinations
5. The Binomial Theorem
6. Advanced Counting Techniques
7. Recurrence Relations
8. Generating Functions
9. Special Counting Numbers and Sequences
10. Graph Theory and Combinatorics
11. Design Theory
12. Probabilistic Combinatorics
13. Algebraic Combinatorics
14. Extremal Combinatorics
15. Applications and Advanced Topics
10.
Graph Theory and Combinatorics
10.1.
Basic Graph Enumeration
10.1.1.
Counting Labeled Graphs
10.1.2.
Counting Unlabeled Graphs
10.1.3.
Polya Enumeration Theory
10.2.
Counting Trees
10.2.1.
Definition of Trees
10.2.2.
Cayley's Formula for Labeled Trees
10.2.3.
Prüfer Sequences
10.2.3.1.
Bijection with Trees
10.2.3.2.
Encoding and Decoding
10.2.4.
Counting Unlabeled Trees
10.2.5.
Rooted Trees
10.2.6.
Binary Trees
10.3.
Counting Spanning Trees
10.3.1.
Definition of Spanning Trees
10.3.2.
The Matrix Tree Theorem
10.3.3.
Kirchhoff's Theorem
10.3.4.
Applications in Network Theory
10.4.
Graph Coloring
10.4.1.
Vertex Coloring
10.4.2.
Chromatic Number
10.4.3.
Chromatic Polynomials
10.4.3.1.
Definition and Properties
10.4.3.2.
Deletion-Contraction Formula
10.4.3.3.
Examples and Calculations
10.4.4.
Edge Coloring
10.4.5.
Applications of Graph Coloring
10.5.
Matchings and Coverings
10.5.1.
Perfect Matchings
10.5.2.
Counting Matchings
10.5.3.
Hall's Marriage Theorem
10.5.4.
König's Theorem
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9. Special Counting Numbers and Sequences
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11. Design Theory